 Open Access
 Authors : Kipkirui Chepkwony, Ch. Ratnam
 Paper ID : IJERTV13IS080048
 Volume & Issue : Volume 13, Issue 08 (August 2024)
 Published (First Online): 24082024
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Analytical and Numerical Investigation of Free Vibration in Beams under Diverse Boundary Conditions and Material Characteristics
Kipkirui Chepkwony
PG Student, Department of Mechanical Engineering, Andhra University, Visakhapatnam
Ch. Ratnam
Professor, Department of Mechanical Engineering, Andhra University, Visakhapatnam
Abstract: Beams are defined by considering their boundary conditions. This paper focused on the Free Vibration Analysis of SiC aluminium reinforced composite beams by considering four boundary conditions
i.e. clampedfree, clampedclamped, clampedsimply supported, and simply supported. The study utilized the EulerBernoulli beam theory to obtain the frequency equation and numerical simulations on the ANSYS Workbench to analyze the free vibration behaviour. The results obtained for SiC aluminiumreinforced composite are compared with those of Aluminium and steel material. The study demonstrates that boundary conditions affect the dynamic response of the composite beams, with clampedclamped boundary conditions yielding higher natural frequencies, followed by clampedsimply supported, simply supported, and clampedfree boundary conditions yielding low natural frequencies. Furthermore, the natural frequencies of SiC/Aluminium composite beams are higher than those of unreinforced Aluminium and steel beams. The study found that the natural frequency of vibrations increases linearly with an increase in the crosssection area of the beam. Finally, the study found that the natural frequency of vibrations increases with an increase in the specific modulus of the material.
Keywords: Natural frequency, Boundary Conditions, SiC/Aluminium composites, Free Vibration Analysis, EulerBernoulli Theory, Finite Element Analysis.

INTRODUCTION
With the advances in technology, composite materials have become the preferred choice for constructing mechanical equipment and structures. Silicon Carbide (SiC) reinforced Aluminium composites are among the best lightweight composites used in high performance applications. These composites are low in density but have high strength and stiffness, making them suitable for applications in the aerospace industry and lightweight structures. In that connection, the study of the vibration characteristics of composite beams is a significant and distinctive area of focus in the field of mechanical engineering. It is particularly essential to quantify the impact of dynamic loading on structures such as tall buildings, long bridges, and industrial machinery. Dynamic loading can lead to fatigue and the initiation of cracks, which are major contributors to accidents and failures in industrial machinery.
Lu et. al. [1] investigated the effect of the vibration frequency on the fatigue of strength of 6061T6 Al Alloys through two stress analysis methods namely nominal and hotspot stress. Mufazzal et al [2] explored the effect of material and surface cracks on the free vibration of the cantilever beam. Agarwallaa and Parhib [3] highlighted that, at the point where cracks appear, the vibration frequency is high. The study was conducted experimentally and with the help of Fine Element software.
Nikhil and Jeyashree [4] investigated the dynamic response of a cracked beam to free vibration. The study utilized ANSYS the effects of cracks at different locations and depths in cantilever beams, fixedfixed beams, and simply supported beams. Mia et.al.[5] studied the natural frequency and mode shapes of transverse vibration on the cracked and uncracked cantilever beams. The analysis was extended to find the impact of crack opening size and mesh refinement. Gawande and More [6] performed free vibration analysis to investigate the effect of the notch on the dynamics of cantilever beams using ANSYS and experiment. The study accounted for the depth and position of the notch in the beam.
Kuppast et al [7] used ANSYS and experimental modelling to investigate the vibration properties of aluminium alloys. The study simulates the effect of increasing copper and silicon content in aluminium alloys. Abdellah et al. [8] investigate the vibration behaviour of aluminium and its alloys. The samples were designed as cantilever plates with and without holes. The analysis was performed with Ansys. Derkach et al. [9] analyzed the effect of the notch on the fundamental mode of vibration for composite cantilever beams using the Finite element analysis.
Quila et al. [10] studied the free vibration analysis of an uncracked and cracked fixed beam using ANSYS. Ferreira and Neto [11] modelled active NiTi filamentreinforced hybrid adaptive composite beams under freefree boundary conditions to study vibration modes and their frequencies. Avcar [12] investigates the free vibration of square crosssectioned Aluminium beams both analytically and numerically under four different boundary conditions. Haskul and Kisa [13] investigate the free vibration of a
doubletapered beam with linearly varying thickness and width using finite element and component mode synthesis methods.
Rossit et al. [14] investigate the vibrational behaviour of Lshaped beams with cracks. The transversal displacements were described using the EulerBernoulli beam theory, while the crack was modelled as an elastically restrained hinge. Wang and Qiao [15] study the vibration behaviour of beams with arbitrary discontinuities and boundary conditions. Charoensuk and Sethaput
[16] performed a vibration analysis experiment and finite element analysis on metal plates with Vnotch at multiple notch locations. Shah et al. [17] used ANSYS to perform the free vibration of composite beams and obtained fundamental natural frequencies. Bozkurt et al. [18] explore analytical approximation techniques in transverse vibration analysis of beams. The computations were performed using the Adomian Decomposition Method (ADM), the Variational Iteration Method (VIM), and the Homotopy Perturbation Method (HPM). Nalbant et al. [19] investigated the free vibration behaviour of stepped nanobeams using the BernoulliEuler theory for beam analysis and Eringen's nonlocal elasticity theory for nanoscale analysis. The system's boundary conditions were defined as simply supported. Teggi [20] explores the free vibration of steel beams under two different boundary conditions: ClampedFree (CF) and ClampedClamped (CC). Santhosh et al. [21] conducted vibration tests on Aluminium 5083 reinforced with varying percentage weights of Silicon Carbide (SiC) and fly ash through experimentation.Bozkurt and Ersoy [22] investigated the vibration behaviour of metal matrix composites (MMCs) used in the aerospace industry
using finite element analysis (FEM). The study focused on AA2124/SiC/25p, a particlereinforced MMC with a homogeneous distribution of particles, hence commonly used in aerospace applications. Acharya et al. [23] analyzed the dynamic characteristics of Aluminium 6061 plates. Modal analysis was performed using both simulation and experimental methods. Kumar et al. [24] conducted a modal analysis of AA5083 composite material reinforced with multiwall carbon nanotubes using analytical and Finite element methods. Taj et al. [25] studied the vibrational characteristics of Aluminium graphite metal matrix composites. The study evaluated the natural frequencies and mode shapes of the composites by experiments and finite element analysis methods. Lakshmikanthan et al. [26] performed the free vibration analysis of A357 Alloy reinforced with dualparticle size Silicon Carbide Metal Matrix composite plates using the Finite Element Method. The study examined the natural frequencies and mode shapes of the composite plates under ClampedClamped and Simply SupportedSimply Supported boundary conditions.
In this paper, free vibration analysis on SiC/Aluminium composite beams will be performed. This study will focus on the effects of the four types of boundary conditions, namely, CF, CC, CSS, and SSSS, on the natural frequencies and mode shapes of the beams. Additionally, the effects of the mechanical properties of the SiC/Aluminium composite on the fundamental natural frequencies of vibration will also be evaluated. These results will be compared to the results of unreinforced aluminium and steel material

ANALYTICAL FORMULATIONS

HalpinTsai equation
Since SiC/Aluminium is a particulate composite, the Halpin Tsai equation predicts the Young Modulus of Elasticity. The equation is as follows:
Em ((1+2sqVp))
(iii). Governing Equation formulations
Let's apply the EulerBernoulli Beam theory to a beam with length L and uniform crosssection. A is considered. Assuming the beam to be elastic with Youngs Modulus E, and the Density .
The relationship between the bending moment and deflection can be expressed as:
EC =
1qVp
( Ep 1)
(1)
M = EI
d2y (4)
dx2
Where q = Em
( Ep +2s)
Em
(2)
Where E is Youngs Modulus, I is the moment of inertia
of the beam and y is the deflection of the beam. For a
EC = Composite Young Modulus , Ep =
Particles Young Modulus , Em = Matrix Young Modulus,
V = Particles Volume s = Particle Aspect ratio (1 2 )
uniform homogenous beam, the equation of motion is obtained as:
EI d4y + d2y = 0, for 0 x L (5)
p , A dx4
dt2

Rule of Mixtures
By application of the rule of mixtures, the density of the composite is obtained as follows:
Where is Density, and A is the crosssection area of the beam.
Then,
c2 d4y + d2y = 0, for 0 x L (6)
= V + V
(3)
dx4 dt2
c p p m m
Where c = EI
(7)
c =Density of composite p=Density of SiC particles
m=Density of Aluminium Matrix
Vp=SiC Particles Volume Vm=Aluminium Matrix Volume
A
The solution of equation (5) is obtained by the method of separation of variables thus, one part depends on position and the other part depends on time.
y = W(x)T(t) (8)
Where W is independent of time and T is independent of position. Substituting equation (8) into equation (6) and simplifying we get,
sinh L sin L cosh L cos L c
0
[ 1 0
cosh L cos L sinh L + sin L ] [c3]=[ ]
(21)
c2 d4W(x) = 1
d2T(t)
(9)
For a nontrivial solution of C1 and C3 then obtaining the
W(x)
dx4
T(t)
dt2
determinant of the coefficients will be zero. Then the
The Equation (9) is expressed as two separate differential equations:
Position variable: d4W 4W(x) = 0 (10)
dx2
solution is as follows:
cos L cosh L = 1 (22)
The first three roots of equation (22) are determined
Where 4 = 2 = A2
(11)
numerically using the MATLAB commands code. The
2
c2 EI
Time variable: d T(x) + 2T(t) = 0 (12)
dt2
The general solution for equation (10) is:
W(x) = C1 sinh x + C2 cosh x + C3 sin x + C4 cos x (13)
C1, C2, C3, and C4 are constants, they are obtained by considering boundary conditions, and sinh and cosh, are
roots L are referred to as eigenvalues.
L = 4.73004 for n = 1, L = 7.85321 for n = 2 ,
L = 10.9956 for n = 3
Where n is the mode number. (23)

ClampedSimply Supported (CSS) beam
The boundary conditions for the CSS beam are;
dw
the hyperbolic functions.
At x = 0, w(x) = 0 and
= 0 (24)
dx
To solve equation (13), we consider the following conditions:

ClampedFree (CF) beam
The boundary conditions are;
dw
At x = L, w(L) = 0 and d2w = 0 (25)
dx2
When the above boundary conditions are considered in
equation (13),
C2 + C3 = 0
By simplifications, the following matrix expression is
At x = 0, w(x) = 0 and
= 0 (14)
dx
obtained
At x = L, d2w = 0 and d3w = 0 (15)
sinh L sin L cosh L cos L c1 0
dx2
dx3
[sinh L + sin L cosh L + cos L] [c2]=[ ]When the above boundary conditions are considered in equation (13),
C1 = 0, C3 = 0
By simplifications, the following matrix expression is
obtained.
0
(26)
For a nontrivial solution of C1 and C2 then obtaining the determinant of the coefficients will be zero. Then the solution is as follows:
tanh L = tan L (27)
[ sinh L + sin L cosh L + cos Lc2 = 0
(16)
The first three roots of equation (27) are determined
cosh L + cos L sinh L sin L ] [c4] [0]
For a nontrivial solution of C2 and C4 then obtaining the determinant of the coefficients will be zero. Then the solution is as follows:
cos L cosh L = 1 (17)
The first three roots of equation (17) are determined numerically using the MATLAB commands code. The roots L are referred to as eigenvalues.
L = 1.87510 for n = 1 L = 4.69409 for n = 2 L = 7.85340 for n = 3 Where n is the mode number. (18)

ClampedClamped (CC) beam
The boundary conditions for the CC beam are;
At x = 0, w(x) = 0 and dw = 0 (19)
dx
At x = L, w(L) = 0 and dw = 0 (20)
numerically using the MATLAB commands code provided in Appendix 1. The roots L are referred to as eigenvalues.
L = 3.9266 for n = 1, L = 7.0686 for n = 2, L = 10.2102 for n = 3 Where n is the mode number. (28)


Simply SupportedSimply Supported (SSSS) beam
The boundary conditions for the SSSS beam are;
At x = 0, w(x) = 0 and d2w = 0 (29)
2
dx2
At x = L, w(L) = 0 and d w = 0 (30)
dx2
When the above boundary conditions are considered in
equation (13),
C1 = 0, C2 = 0
By simplifications, the following matrix expression is
obtained
dx sinh L sin L
c3 0
When the above boundary conditions are considered in
[sinh L sin L] [c ]=[] (31)
equation (13), 4 0
C2 = 0, C4 = 0
By simplifications, the following matrix expression is
obtained.
For a nontrivial solution of C3 and C4 then obtaining the
determinant of the coefficients will be zero. Then the solution is as follows:
sinLsinhL = 0 (32)
The first three roots of equation (32) are determined numerically using the MATLAB commands code provided in Appendix 1. The roots L are referred to as eigenvalues.
L = 3.14159 for n = 1, L = 6.28318 for n = 2,
L = 9.42478 for n = 3 Where n is the mode number. (33)
Equations 17, 22, 27, and 32 are called frequency equations. By rearranging equation 11, it can be expressed as follows:


RESULTS AND DISCUSSION

Natural frequency across the material:
SiC Particles and Aluminium material properties were adapted from Yuan et al. [27]. To obtain the Elastic Modulus (Ec) of the composite, the rule of mixtures is applied with the help of the HalpinTsai equation.
(440 1)
q = 70 = 0.5692 = 96.14 Gpa (440 + 2(1.5))
70
n n 4
= ( L)2 EI
AL
, Where n=1,2, 3. n modes
EC =
70×109 ((1 + 2(1.5)(0.5692)(0.15)))
1 ((0.5692)(0.15))
numbers. (34)
Table 1 Properties of materials
c = pVp + mVm,
c = (3210×0.15) + (2700×0.5) = 2777 Kg/m3
Properties
SiC Particles
Aluminium
Steel
Density (kg/m3)
3210
2700
7850
Young
Modulus x 109 Pa
440
70
210
Particle Volume (Vp) in Percentag
e (%)
15
85
–
Aspect
ratio of particles
12
–
–
Problem: To demonstrate the vibration analysis of the
beam, the model with the following dimensional characteristics is considered for evaluation: Length (L) = 500mm, width (b) = 50mm, depth (d) = 10mm.
Table 2 Natural Frequency of ClampedFree (CF) beam
Mode
Method
SiC/Aluminium Composite
Aluminium
Steel
Natural Frequency f in Hz
Mode 1
Analytical
38.02
32.90
33.42
Ansys
38.45
32.91
33.65
Mode 2
Analytical
238.29
206.17
209.43
Ansys
240.51
205.86
210.51
Mode 3
Analytical
666.98
577.08
586.20
Ansys
672.11
575.21
588.19
Table 3 Natural Frequency of ClampedClamped (CC) beam
Mode
Method
SiC/Aluminium Composite
Aluminium
Steel
Natural Frequency f in Hz
Mode 1
Analytical
241.95
209.34
212.65
Ansys
246.73
210.74
215.05
Mode 2
Analytical
666.94
577.05
586.17
Ansys
678.04
579.12
590.95
Mode 3
Analytical
1307.47
1131.25
1149.13
Ansys
1324.60
1131.30
1154.43
Table 4 Natural Frequency of ClampedSimply Supported (CSS) beam
Mode
Method
SiC/Aluminium Composite
Aluminium
Steel
Natural Frequency f in Hz
Mode 1
Analytical
166.74
144.26
146.54
Ansys
168.40
145.40
147.37
Mode 2
Analytical
540.33
467.51
474.89
Ansys
544.72
470.78
477.71
Mode 3
Analytical
1127.36
975.42
990.83
Ansys
1133.80
978.74
992.08
Table 5 Natural Frequency of Simply SupportedSimply Supported (SSSS) beam
Mode
Method
SiC/Aluminium Composite
Aluminium
Steel
Natural Frequency f in Hz
Mode 1
Analytical
109.92
92.35
93.81
Ansys
106.71
92.33
93.79
Mode 2
Analytical
426.92
369.39
375.22
Ansys
426.56
369.03
374.85
Mode 3
Analytical
960.59
832.12
844.25
Ansys
956.60
829.11
842.18
SiC/Aluminium Composite presented the highest Natural frequency in all the modes that have been considered between Aluminium and Steel. This is due to the higher stiffness and lower density of SiC/Aluminium composite hence natural frequencies of the structure occur at higher values. Aluminium is given higher stiffness by reinforcing with SiC particles and this in turn improves the vibrational behavior of the composite. Natural frequencies for Aluminium are found to be less than those of SiC/Aluminium composite but close to that of Steel.
Aluminium has been found to have a lower density than Steel but the modulus of elasticity is lower; this results in Aluminium and Steel materials having similar natural frequencies. Natural frequencies of Steel are slightly higher than that of the Aluminium across all modes. However, a comparison between Aluminium and Steel shows that the two are not very different, and hence have very close values of the vibrational frequency.

ANSYS Graphical results for SiC/Aluminium beam
Figure 1 CF first three Mode Shapes for SiC/Aluminium beam.
Figure 2 CC first Mode Shapes for SiC/Aluminium beam.
Figure 4 SSSS first Mode shapes for SiC/Aluminium beam.
Figure 3 CSS first Mode shapes for SiC/Aluminium beam.

Comparative analysis between Analytical and ANSYS Results Results for SiC/Aluminium Composites are considered for clampedfree beam as an example.
The differences in results obtained by the two methods are expressed in percentages. These percentages are obtained as follows. If the three modes of vibration n = 1,2,3 , analytical natural frequency as fn analyticl and Ansys natural frequency as fn ansys, then:
fn analyticlfn ansys
and modes. These variations include the analytical approximations made during analysis and would fall below 1%. Comparing Mode 1 for the SSSS beam, it is safe to say that the SiC/Aluminium composite diverged most (3.02 Hz or about 2.9% deviation) from the actual, probably due to some difficulties in accurately simulating the composite
Percentage deviation = (
fn analyticl
) Ã— 100%
material. For modes 2 and 3, the difference in the analytical solution and the Ansys solution is lower for higher
The Analytical and Ansys natural frequencies differ by – 1.12%, 0.93%, and 0.77% under CF boundary conditions
beam, 1.98%, 1.66%, and 1.31% under CC boundary
conditions, 1.00%, 0.81% and 1.31% under CSS
boundary conditions and 2.92%, 0.08% and 0.42% under SSSS boundary conditions.
SSSS
CSS
CC
CF
3.00%
2.00%
1.00%
0.00%
1.00%
2.00%
3.00%
4.00%
Mode 3 Mode 2 Mode 1
Figure 5 shows the close conformity of solutions obtained by analytical and the Ansys approaches for all the materials
frequency modes indicating that the models are more accurate at higher modes. This could be because higher modes are less sensitive to the boundary condition.
Figure 5 Percentage deviation in natural frequency obtained by Analytical and Ansys.

Effects of specific modulus on natural frequencies of the beam.
The properties of the material determine the basic associated frequencies of vibration of beams. The beam material has a unique property called Specific Modulus (E/) which has to be considered
Properties of Materials
SiC/Aluminium Ec
= 96.14 Gpa, = 2777kg/m3 , E = 34.62x106m2/s2
Aluminium E = 70 Gpa, = 2700kg/m3, E = 25.93x106m2/s2
Steel E = 210 Gpa, = 7850kg/m3, E = 26.75x106m2/s2
Table 7 Natural frequencies versus boundary conditions at specific modulus of materials.
Specific Modulus
Boundary Condition
Mode
25.93
26.75
36.62
Natural Frequency in Hz
CF
Mode 1
32.9
33.42
38.02
Mode 2
206.17
209.43
238.29
Mode 3
577.08
586.2
666.98
CC
Mode 1
209.34
212.65
241.95
Mode 2
577.05
586.17
666.94
Mode 3
1131.25
1149.13
1307.47
CSS
Mode 1
144.26
146.54
166.74
Mode 2
467.51
474.89
540.33
Mode 3
975.42
990.83
1127.36
SSSS
Mode 1
92.35
93.81
109.92
Mode 2
369.39
375.22
426.92
Mode 3
832.12
844.25
960.59
Analytical results from Tables 2 to 5 are used to generate Table 7 above. The data in Table 7 are used to generate Figure 9 below. The specific modulus is one of those parameters which determine the natural frequency of a given material. Generally, a higher value of specific modulus results in higher natural frequencies, because the material is stiffer or the structure is lighter. SiC/Al (Specific Modulus = 36.62×106) used in the present study exhibits the highest natural frequencies across all the boundary conditions and modes. Consequently, the higher specific
modulus of the SiC/Al means higher stiffness resulting in higher resistance to deformation and thus, higher natural frequencies. Steel (Specific Modulus = 26.75 x 106) exhibits natural frequencies a little higher than Aluminium, but lower as compared to SiC/Al. Steel material has a higher density compared to aluminium and a relatively higher elastic modulus and therefore natural frequencies. Aluminium (Specific Modulus = 25.93 x 106) has the lowest specific modulus among the three materials resulting in the lowest natural frequencies. Due to it having a lower stiffness
the material can undergo larger deformation than the other two materials, which in turn lowers natural frequencies.
From Figure 9, it was noted that the natural frequency increases with an increase in the Specific Modulus of the material. The rate of increase in natural frequency is more pronounced in the CC mode 3 condition, followed by the CSS mode 3 condition, and SSSS mode 3 condition. The intermediate increase was noted at CC mode 2 and CF mode 3 conditions, followed by CSS mode 2 condition, and SSSS mode 2 condition. The low increase was noted at C C mode 1 and CF mode 2 condition, followed by CSS
mode 1 condition, SSSS mode 1 condition and CF mode 1 condition.
The frequency curve of the beam at CF mode 1 condition can be noted to be a horizontal line. This signifies that the effect of material properties on the natural frequency of the CF mode 1 is insignificant. This observation portrayed the effect of boundary conditions on the vibration of the beam. The boundary condition does offer a different stiffness effect to the beam; thus, the free end of the CF beam lowers the stiffness, hence in result lowers the natural frequencies of vibration.
CF Mode 1
CC Mode 2
CF Mode 2
CC Mode 3
CF Mode 3
CSS Mode 1
CC Mode 1
CSS Mode 2
1400
CSS Mode 3
SSSS Mode 1
SSSS Mode 2
SSSS Mode 3
CC Mode 3
1200
CSS Mode 3
1000
SSSS Mode 3
800
CC Mode 2 & CF Mode
3
600
CSS Modee 2
400
SSSS Mode 2
CSS Mode 1 and CF
Mode 2
200
CC Mode 1
SSSS Mode 1
0
CF Mode 1
25.93
26.75
Specific Modulus X 106
36.62
Natural Frequency In Hz
Figure 9 Natural frequencies versus Specific Modulus.

Effects of Crosssection area of the beam on the natural frequencies of vibration.
Effects of boundary conditions – Results for SiC/Aluminium Composites are considered as an example.
The results for SiC/Aluminium are extracted from Tables 2 to 5 and populated as shown in Table 6. For the CF boundary condition natural frequencies are lowest compared to CC, CSS, and SSSS across all modes. Thus,
the CF condition provides more displacement at the free end resulting in low stiffness and consequently low natural frequencies.
Table 6 Natural frequencies of SiC/Aluminium beam supported by different boundary conditions.
Boundary Conditions
Analysis Method
Natural Frequency f in Hz for SiC/Aluminium Composite Beam
Mode 1
CF
CC
CSS
SSSS
Analytical
38.02
241.95
166.74
109.92
Ansys
38.445
246.73
168.4
106.71
Mode 2
Analytical
238.29
666.94
540.33
426.92
Ansys
240.51
678.04
544.72
426.56
Mode 3
Analytical
666.98
1307.47
1127.36
960.59
Ansys
672.11
1324.60
1133.80
956.60
Mode 1
400
200
0
CF
CC
CSS
SSSS
Analytical
Ansys
The CC boundary condition provided the highest natural frequency across all modes. This condition provides much no freedom of movement of the beam hence resulting in high stiffness and high natural frequencies. The CSS boundary condition resulted in natural frequencies higher than CF and SSSS but lower than CC conditions. The C SS has one end restraint and the other end is free to rotate.
These conditions provide intermediate natural frequencies. The SSSS boundary conditions result in natural frequencies lower than CSS and higher than CF. This condition also permits some extent of rotation at the supports which results in a lower degree of stiffness as compared to CC. Figures 6, 7, and 8 provide graphical representations of the impact of boundary conditions on the natural frequencies of beams.
Mode2
1000
0
CF
CC
CSS
SSSS
Analytical
Ansys
atural Frequency
in Hz
atural Frequency in
Hz
Figure 6 Mode 1 natural frequencies versus boundary conditions.
Figure 7 Mode 2 natural frequencies versus boundary conditions
Mode 3
1500
1000
500
0
CF
CC
CSS
SSSS
Analytical
Ansys
Natural Frequency in Hz
Figure 8 Mode 3 natural frequencies versus boundary conditions.
For the presentation of this study, a beam of the following characteristics was considered. The Length = 500mm, Crosssection area A1=0.0005m2, A2= 0.0008m2, A3= 0.0015m2. The Physical properties of SiC/Aluminium composite are Density = 2777kg/m3, and Estimated Young Modulus of Elasticity = 96.14 GPa.
Table 8 Natural frequency for Beam under four different boundary conditions versus crossarea
Boundary conditions
Mode n of Vibration
Natural Frequency in Hz
A1
A2
A3
CF
Mode 1
38.02
76.04
114.06
Mode 2
238.29
476.56
714.79
Mode 3
666.98
1333.91
2000.74
CC
Mode 1
241.95
483.88
725.78
Mode 2
666.94
1333.84
2000.64
Mode 3
1307.47
2614.86
3922.05
CSS
Mode 1
166.74
333.46
500.16
Mode 2
540.33
1080.63
1620.85
Mode 3
1127.36
2254.65
3381.76
SSSS
Mode 1
109.92
213.46
320.17
Mode 2
426.92
853.83
1280.66
Mode 3
960.59
1921.11
2881.49
From the consideration, the beam has a fixed length (L) and material properties, thus, Figure 10 depicts the natural frequency of vibrations to increase linearly with an increase in crosssection area. The rate of an increase in natural frequency is more pronounced for the CC boundary condition, followed by CSS, SSSS, and CF. These effects happen because the mode shape constant for CC and C SS are higher compared to SSSS and CF boundary conditions. This underscores the role boundary conditions play as the CC condition yields the highest frequencies due to maximum stiffness and the CF condition yields the lowest due to greater flexibility.


CONCLUSIONS
The natural frequencies for the beam under four different boundary conditions were estimated analytically and numerically using ANSYS Workbench. The results obtained were consistently in agreement for both methods. It was noted that higher natural frequencies were achieved by SiC/ Aluminium composite beams across all four boundary conditions considered, followed by structural Steel and Aluminium beams. This is because SiC/ Aluminium composites have a higher specific modulus than Steel and Aluminium.
The higher natural frequency is experienced in CC boundary conditions, followed by CSS, SSSS, and lower in CF boundary conditions. The linear increase in natural frequencies is depicted to increase with the beam crosssectional area when the horizontal length
and mass distribution of the beam are constant. The rate of increase in the natural frequency is more pronounced in CC boundary conditions, followed by CSS, SSSS, and least under CF boundary conditions.
CF Mode 1
CC Mode 2
CSS Mode 3
CF Mode 2
CC Mode 3
SSSS Mode 1
CF Mode 3
CSS Mode 1
SSSS Mode 2
CC Mode 1
CSS Mode 2
SSSS Mode 3
4500
4000
CC Mode 3
CSS Mode 3
3500
SSSS Mode 3
3000
CC Mode 1 & CF
Mode 2
2500
SSSS Mode 1
CC Mode 2 & CF
Mode 3
2000
CSS Mode 2
1500
SSSS Mode 2
1000
500
CSS Mode 1
CF Mode 1
0
A1
A2
A3
Crosssection Area
Natural Frequency In Hz
Figure 10 Natural frequency versus crosssection area at different boundary conditions.
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